## Trace, Determinant, and Eigenvalue (Harvard University Exam Problem)

## Problem 389

**(a)** A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$.

Find $\det(A)$.

**(b)** A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$.

**(c)** A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of $A$?

(*Harvard University, Linear Algebra Exam Problem*)